Some common exercises presented in introductory acoustics courses and texts illustrate solutions involving eigenvalues and eigenfunctions. Challenging extensions of these, even for one-dimensional (1D) systems, might involve a mass or spring loading the acoustic medium at an end point or at an interior point. These problems might be extended further by requiring that some given function be expanded in a series of the eigenfunctions, but such extended problems may lead to unexpected complications in regard to eigenfunction orthogonality. In this paper, Sturm-Liouville theory is used to develop a systematic method for predetermining eigenfunction orthogonality for 1D systems loaded at end points or interior points or having properties that change with jump discontinuities.

中文翻译：

---

具有内部或端点条件的一维声学系统的本征函数正交性。

声学入门课程和课文中介绍的一些常见练习说明了涉及特征值和特征函数的解决方案。即使对于一维（1D）系统，这些扩展的挑战性扩展也可能涉及在端点或内部点对声学介质进行质量或弹簧加载。通过要求在一系列本征函数中扩展某些给定的函数，可以进一步扩展这些问题，但是这种扩展的问题可能会导致本征函数正交性方面的意外复杂化。在本文中，Sturm-Liouville理论用于开发一种系统的方法，该方法可以预先确定一维系统在端点或内部点处加载的特征函数正交性，或者具有随跳跃不连续性而变化的特性。